Group #22616052 — subgroups

Subgroups — #22616052

Order 9344.

448 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 165 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14556591	C₂	1	1	yes	maximal, characteristic
2	#17988517	C₂	1	4	yes	maximal
3	#17988507	C₂	1	4	yes	maximal
4	#17988456	C₂	1	2	yes	maximal
5	#17988452	C₂	1	2	yes	maximal
6	#17988433	C₂	1	1	yes	maximal, characteristic
7	C₂×#125×C₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₄₆	C₂²	1	2	yes	—
9	#14556591	—	2	4	no	—
10	C₂×C₄⋊Dic₇₃	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₇₃	C₂²	1	2	yes	—
12	C₂×#11924007	C₂²	1	8	yes	—
13	C₂×C₄×Dic₇₃	C₂²	1	2	yes	—
14	#14556498	C₂²	1	2	yes	—
15	#14556498	C₂²	1	2	yes	—
16	#14556498	C₂²	1	8	yes	—
17	#125×C₇₃	—	2	4	no	—
18	C₂×C₄⋊C₄×C₇₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₇₃	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₇₃	C₂²	1	2	yes	—
21	C₂×C₄⋊C₄×C₇₃	C₂²	1	2	yes	—
22	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₄₆	—	2	4	no	—
24	C₂×Dic₁₄₆	—	4	4	no	—
25	C₂²×Dic₇₃	C₂³	1	4	yes	—
26	C₂²×Dic₇₃	C₂³	1	4	yes	—
27	C₄⋊Dic₇₃	—	2	4	no	—
28	C₄⋊Dic₇₃	—	2	4	no	—
29	#11924007	—	4	8	no	—
30	C₄×Dic₇₃	Q₈	1	8	yes	complemented
31	C₂²×C₄×C₇₃	C₂³	1	2	yes	—
32	C₂²×C₄×C₇₃	C₂³	1	2	yes	—
33	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
34	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₇₃	—	2	4	no	—
37	C₄⋊C₄×C₇₃	—	2	4	no	—
38	C₄⋊C₄×C₇₃	—	2	2	no	—
39	C₄⋊C₄×C₇₃	—	2	2	no	—
40	C₄²×C₇₃	D₄	1	4	yes	—
41	C₂³×C₇₃	C₂⁴	1	1	yes	characteristic
42	C₂×C₄×C₇₃	—	2	2	no	—
43	C₂×C₄×C₇₃	—	2	2	no	—
44	C₂×C₄×C₇₃	—	2	2	no	—
45	C₂×C₄×C₇₃	—	2	2	no	—
46	C₂×C₄×C₇₃	—	2	4	no	—
47	C₂×C₄×C₇₃	—	2	4	no	—
48	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₇₃	C₂×Q₈	1	4	yes	—
52	C₂×Dic₇₃	—	2	4	no	—
53	C₂×Dic₇₃	—	2	4	no	—
54	C₂×Dic₇₃	—	2	4	no	—
55	C₂×Dic₇₃	—	2	8	no	—
56	C₂×Dic₇₃	C₂×Q₈	1	8	yes	—
57	Dic₁₄₆	—	4	8	no	—
58	C₂²×C₇₃	C₂²×D₄	1	1	yes	characteristic
59	C₂²×C₇₃	C₂²×Q₈	1	2	yes	—
60	C₂²×C₇₃	H(5,𝔽₂)	1	2	yes	—
61	C₂²×C₇₃	2₋¹⁺⁴	1	2	yes	—
62	C₄×C₇₃	—	2	2	no	—
63	C₄×C₇₃	—	2	2	no	—
64	C₄×C₇₃	—	4	2	no	—
65	C₄×C₇₃	—	4	2	no	—
66	C₄×C₇₃	C₄⋊Q₈	1	4	yes	—
67	Dic₇₃	—	2	8	no	—
68	Dic₇₃	—	4	4	no	—
69	C₂×C₇₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
70	C₂×C₇₃	#535	1	1	yes	characteristic
71	C₂×C₇₃	#536	1	1	yes	characteristic
72	C₂×C₇₃	#544	1	4	yes	—
73	#2660	—	73	1	no	maximal, Sylow
74	C₇₃	#2660	1	1	yes	characteristic, complemented, Sylow
75	C₂×C₄⋊Q₈	—	73	1	no	—
76	C₂×#125	D₇₃	1	1	yes	characteristic
77	#398	—	73	4	no	—
78	#395	—	73	4	no	—
79	#384	—	73	2	no	—
80	#382	—	73	2	no	—
81	#376	—	73	1	no	—
82	C₂²×Q₈	—	73	2	no	—
83	C₄⋊Q₈	—	146	4	no	—
84	#125	—	2	4	no	—
85	C₂×C₄⋊C₄	—	73	8	no	—
86	C₂×C₄⋊C₄	—	73	2	no	—
87	C₂×C₄⋊C₄	—	73	2	no	—
88	C₂×C₄⋊C₄	C₂×D₇₃	1	1	yes	characteristic
89	C₂×C₄⋊C₄	C₂×D₇₃	1	1	yes	characteristic
90	C₂×C₄⋊C₄	C₂×D₇₃	1	2	yes	—
91	C₂×C₄⋊C₄	C₂×D₇₃	1	2	yes	—
92	C₂×C₄²	—	73	2	no	—
93	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
94	#95	—	73	2	no	—
95	#95	—	73	2	no	—
96	#95	—	73	8	no	—
97	C₂×Q₈	—	292	4	no	—
98	C₂×Q₈	—	146	4	no	—
99	C₂²×C₄	—	73	4	no	—
100	C₂²×C₄	—	73	4	no	—
101	C₂²×C₄	C₂²×D₇₃	1	2	yes	—
102	C₂²×C₄	C₂²×D₇₃	1	2	yes	—
103	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
104	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
105	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
106	C₄⋊C₄	—	2	4	no	—
107	C₄⋊C₄	—	2	4	no	—
108	C₄⋊C₄	—	2	2	no	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	292	8	no	—
111	C₄⋊C₄	—	146	4	no	—
112	C₄⋊C₄	—	146	4	no	—
113	C₄²	—	73	8	no	—
114	C₄²	C₂²⋊D₇₃	1	4	yes	—
115	C₂³	C₂³×D₇₃	1	1	yes	characteristic
116	Q₈	—	292	8	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	73	8	no	—
124	C₂×C₄	—	146	4	no	—
125	C₂×C₄	—	146	8	no	—
126	C₂×C₄	—	146	4	no	—
127	C₂×C₄	—	146	4	no	—
128	C₂×C₄	C₂×C₂²⋊D₇₃	1	2	yes	—
129	C₂×C₄	C₂×C₂²⋊D₇₃	1	2	yes	—
130	C₂×C₄	C₂×C₂²⋊D₇₃	1	2	yes	—
131	C₂×C₄	Q₈×D₇₃	1	4	yes	—
132	C₂²	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
133	C₂²	C₂×Q₈×D₇₃	1	2	yes	—
134	C₂²	#14556637	1	2	yes	—
135	C₂²	#14556638	1	2	yes	—
136	C₄	—	2	2	no	—
137	C₄	—	2	2	no	—
138	C₄	—	4	2	no	—
139	C₄	—	4	2	no	—
140	C₄	—	292	4	no	—
141	C₄	—	146	8	no	—
142	C₄	#14556589	1	4	yes	—
143	C₂	C₂×#14556589	1	1	yes	characteristic
144	C₂	#17989151	1	1	yes	characteristic
145	C₂	#17989158	1	1	yes	characteristic
146	C₂	#17989217	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556591

C₂

yes

maximal, characteristic

#17988517

C₂

yes

maximal

#17988507

C₂

yes

maximal

#17988456

C₂

yes

maximal

#17988452

C₂

yes

maximal

#17988433

C₂

yes

maximal, characteristic

C₂×#125×C₇₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₄₆

C₂²

yes

—

#14556591

—

C₂×C₄⋊Dic₇₃

C₂²

yes

—

C₂×C₄⋊Dic₇₃

C₂²

yes

—

C₂×#11924007

C₂²

yes

—

C₂×C₄×Dic₇₃

C₂²

yes

—

#14556498

C₂²

yes

—

#14556498

C₂²

yes

—

#14556498

C₂²

yes

—

#125×C₇₃

—

C₂×C₄⋊C₄×C₇₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₃

C₂²

yes

—

C₂×C₄⋊C₄×C₇₃

C₂²

yes

—

C₂×C₄²×C₇₃

C₂²

yes

characteristic

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂²×Dic₇₃

C₂³

yes

—

C₂²×Dic₇₃

C₂³

yes

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924007

—

C₄×Dic₇₃

Q₈

yes

complemented

C₂²×C₄×C₇₃

C₂³

yes

—

C₂²×C₄×C₇₃

C₂³

yes

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

D₄

yes

—

C₂³×C₇₃

C₂⁴

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂×Q₈

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂×Q₈

yes

—

Dic₁₄₆

—

C₂²×C₇₃

C₂²×D₄

yes

characteristic

C₂²×C₇₃

C₂²×Q₈

yes

—

C₂²×C₇₃

H(5,𝔽₂)

yes

—

C₂²×C₇₃

2₋¹⁺⁴

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₄⋊Q₈

yes

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

C₂×C₄⋊Q₈

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

#2660

—

maximal, Sylow

C₇₃

#2660

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×#125

D₇₃

yes

—

—

—

—

—

C₂²×Q₈

—

C₄⋊Q₈

—

146

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇₃

yes

characteristic

C₂×C₄⋊C₄

C₂×D₇₃

yes

characteristic

C₂×C₄⋊C₄

C₂×D₇₃

yes

—

C₂×C₄⋊C₄

C₂×D₇₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₇₃

yes

—

—

—

C₂×Q₈

—

292

—

C₂×Q₈

—

146

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

C₂²×D₇₃

yes

—

102

C₂²×C₄

C₂²×D₇₃

yes

—

103

C₂²×C₄

C₂²×D₇₃

yes

characteristic

104

C₂²×C₄

C₂²×D₇₃

yes

characteristic

105

C₂²×C₄

C₂²×D₇₃

yes

characteristic

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

292

—

111

C₄⋊C₄

—

146

—

112

C₄⋊C₄

—

146

—

113

C₄²

—

114

C₄²

C₂²⋊D₇₃

yes

—

115

C₂³

C₂³×D₇₃

yes

characteristic

116

Q₈

—

292

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

146

—

125

C₂×C₄

—

146

—

126

C₂×C₄

—

146

—

127

C₂×C₄

—

146

—

128

C₂×C₄

C₂×C₂²⋊D₇₃

yes

—

129

C₂×C₄

C₂×C₂²⋊D₇₃

yes

—

130

C₂×C₄

C₂×C₂²⋊D₇₃

yes

—

131

C₂×C₄

Q₈×D₇₃

yes

—

132

C₂²

C₂²×C₂²⋊D₇₃

yes

characteristic

133

C₂²

C₂×Q₈×D₇₃

yes

—

134

C₂²

#14556637

yes

—

135

C₂²

#14556638

yes

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

292

—

141

C₄

—

146

—

142

C₄

#14556589

yes

—

143

C₂

C₂×#14556589

yes

characteristic

144

C₂

#17989151

yes

characteristic

145

C₂

#17989158

yes

characteristic

146

C₂

#17989217

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇₃
commutator, socle	C₂³×C₇₃	C₂⁴
Fitting	C₂×#125×C₇₃	C₂
hypercenter	C₂×#125	D₇₃
hypocenter	C₇₃	#2660
LCS	G ▷ C₂³×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×#125

subgroup

G/H

center, Frattini

C₂³

C₂³×D₇₃

commutator, socle

C₂³×C₇₃

C₂⁴

Fitting

C₂×#125×C₇₃

C₂

D₇₃

C₇₃

G ▷ C₂³×C₇₃ ▷ C₇₃

C₁ ◁ C₂³ ◁ C₂×#125

N	H ≅ G/N	type
30. C₄×Dic₇₃	Q₈	semidirect
74. C₇₃	#2660	semidirect

H ≅ G/N

type

30. C₄×Dic₇₃

Q₈

semidirect

74. C₇₃

#2660

semidirect

Subgroups — #22616052

Conjugacy classes of subgroups

Complements